We don't always understand why God allows some things to happen; we wish we could, but in all honesty they remain a mystery to us. Some day, when we get to heaven, we will understand -- but until then, all we can do is learn to trust God, even in the dark places of life. As the Bible says, "We live by faith, not by sight" (2 Corinthians 5:7).
But no -- God does not make mistakes. That doesn't mean He is directly responsible for everything that happens in the world; evil is real, and some things happen in spite of God's perfect will. But God is perfect in all He does -- because He is God, and God is perfect. He isn't like we are, sinful and limited and imperfect; He is absolutely without sin.
And because He is perfect, we can trust Him in every situation we will ever face -- without exception. The Bible says, "As for God, his way is perfect; the word of the Lord is flawless. He is a shield for all who take refuge in him" (Psalm 18:30). If you have never done so, make Him your "shield" against the evils of this world by opening your heart and life to Jesus Christ.
Yes, this world is filled with heartache and sorrow. But it won't always be that way! Some day, Christ will come again, and death and sin and Satan will be banished. Put your faith and trust in Him, for He alone gives us hope -- both for this life, and for eternity.